Ken Ward's Mathematics Pages Vieta's Root Formulas

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Vieta's Root Formulas

Vieta's formulas relate the coefficients of a polynomial to its roots. François
Viète (or Vieta), seigneur de la
Bigotière, generally known as Franciscus Vieta (1540-1603),
was a French mathematician. Naturally, there is some confusion over
which name to use!

This page is concerned with the use of Vieta's formulas to examine
polynomials and does not deal with the theory. The formulas are sometimes called Viète's formulas or
relations. If the coefficient of xn isn't 1, then we can divide by the
coefficient of xn (a_{n}) An equation: x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}...
+a_{0}=0 [1] can be written: (x-r_{1})(x-r_{2})...(x-r_{n})=0
[2] Where r_{1}, r_{2},
etc are the n roots of the equation [1] (This is based on the
Fundamental Theorem of Algebra) By observation, we can see that the sum
of the roots is equal to -a_{n-1}

If n is even, the product is equal to
the constant term. If n is odd, the product
is equal to the negative
of the constant term.

The sum of the roots is:

And the product is:

Quadratic Formulas

A quadratic equation with roots α and β can be written:
ax^{2}+bx+c=a(x-α)(x-β)=0
For which, Vieta's root forumula is:

Cubic Formulas

For the cubic equation:
ax^{3}+bx^{2}+cx+d=a(x-α)(x-β)(x-γ)=0

Quartic Formulas

For the quartic equation:
ax^{4}+bx^{3}+cx^{2}+dx+e-a(x-α)(x-β)(x-γ)(x-δ)=0

^{}

Example 1

x^{2}+x-6
If the roots are α and β then, according to Vieta:

α + β=-1 (sum of roots equal to minus coefficient of n-1)
αβ=-6 (product is equal to the constant term, because n is even)
The roots are -3 and 2, so:
-3+2=-1
(-3)2=-6

Example 2

x^{3}+2x^{2}-5x-6
If the roots are α, β and γ, then according to Vieta:

α+β+γ=-2 (sum of roots equal to minus coefficient of n-1, that is (-1)*2)
αβγ=6 (product is equal to minus the constant term (-1)*(-6), because n is odd)

The roots are -3, 2 and -1, so:
-3+2-1=2
(-3)2(-1)=6

Example 3

x^{4}-2x^{3}-13x^{2}+14x+24
If the roots are α, β, γ and δ, then according to Vieta:

α+β+γ+δ=2 (sum of roots equal to minus coefficient of n-1, that is (-1)*(-2))
αβγδ=24 (product is equal to minus the constant term, because n is even)
The roots are -3, 2 -1 and 4 so:
-3+2-1+4=2
(-3)2(-1)(4)=24

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